What is the integral of x^2e^{-x/y} ?

The integral of x^2e^{-x/y} is -ye^{-x/y}x^2+2y(-ye^{-x/y}x-y^2e^{-x/y})+C

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integral of x^2e^{-x/y}

\int x^{2} e^{- \frac{x}{y}} d x

- y e^{- \frac{x}{y}} x^{2} + 2 y (- y e^{- \frac{x}{y}} x - y^{2} e^{- \frac{x}{y}}) + C

Solution steps

\int x^{2} e^{- \frac{x}{y}} d x

Apply Integration By Parts

= - y e^{- \frac{x}{y}} x^{2} - \int - 2 y e^{- \frac{x}{y}} x d x

\int - 2 y e^{- \frac{x}{y}} x d x = - 2 y (- y e^{- \frac{x}{y}} x - y^{2} e^{- \frac{x}{y}})

= - y e^{- \frac{x}{y}} x^{2} - (- 2 y (- y e^{- \frac{x}{y}} x - y^{2} e^{- \frac{x}{y}}))

Simplify

= - y e^{- \frac{x}{y}} x^{2} + 2 y (- y e^{- \frac{x}{y}} x - y^{2} e^{- \frac{x}{y}})

Add a constant to the solution

= - y e^{- \frac{x}{y}} x^{2} + 2 y (- y e^{- \frac{x}{y}} x - y^{2} e^{- \frac{x}{y}}) + C

What is the integral of x^2e^{-x/y} ?
The integral of x^2e^{-x/y} is -ye^{-x/y}x^2+2y(-ye^{-x/y}x-y^2e^{-x/y})+C